Independence and Connectivity of Connected Domination Critical Graphs

A graph $G$ is said to be $k$-${\gamma }_c$-critical if the connected domination number ${\gamma }_c(G)=k$ and ${\gamma }_c(G+uv)<k$ for every $uv\in E(\bar{G})$. Let $\delta ,\kappa$ and $\alpha$ be respectively the minimum degree, the connectivity and the independence number. In this paper, we show that a $3$-${\gamma }_c$-critical graph $G$ satisfies $\alpha \leq \kappa + 2$\mathrm{.}Moreover,if$\kappa \geq 3$,then$\alpha = \kappa + p$ifandonlyif$\alpha = \delta + p$forall$p \in {1, 2}$\mathrm{.}Weshowthatthecondition$\kappa + 1 \leq \alpha \leq \kappa + 2$isbestpossibletoprovethat$\kappa = \delta$. By these result, we conclude our paper with an open problem on Hamiltonian connected of $3$-${\gamma }_c$-critical graphs.